Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C3×#1783 ⋊ C2 |
| Sylow | C3 ⋊ #23897 |
1952 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 1952 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^8,3|f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,i^a=i^2,c^b=cg,e^c=eh,f^c=fh,i^c=i^2,f^d=fh72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#23897, #72231720, #72231726, … |
| maximal normal subgroup of | D3×#23897, #72231720, #72231726, … |
| maximal quotient of | C2×#10826749, #72252300, #72270291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 201130);